Strichartz estimates to fractional Schr\"odinger equations

Abstract

In this paper, we firstly study Strichartz estimates $\|e^{it D^\alpha} u_0\|_{L_t^q(\mathbb{R};L_x^r(\mathbb{R}^d))}\leq C(d,\alpha,q,r,s)\|u_0\|_{\dot{H}^s}.$ We show some counterexamples for $(q,r) = (2,\infty)$. Then we consider the embedding $X^{s,b}_\alpha \hookrightarrow L_t^q(\mathbb{R};L_x^r(\mathbb{R}^d))$, where $\|u\|_{X_\alpha^{s,b}}:=\|\langle\xi\rangle^s\langle \tau-|\xi|^\alpha\rangle^b\hat{u}(\tau,\xi)\|_{L^2_{\tau,\xi}}$. We present the necessary and sufficient conditions for this embedding.